IZA DP No. 3477

Hyperbolic Discounting and the Phillips Curve

Liam GrahamDennis J. Snower

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Forschungsinstitutzur Zukunft der ArbeitInstitute for the Studyof Labor

April 2008

Hyperbolic Discounting and the

Phillips Curve

Liam Graham University College London

Dennis J. Snower

Kiel Institute for the World Economy, University of Kiel and IZA

Discussion Paper No. 3477 April 2008

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IZA Discussion Paper No. 3477 April 2008

ABSTRACT

Hyperbolic Discounting and the Phillips Curve Using a standard dynamic general equilibrium model, we show that the interaction of staggered nominal contracts with hyperbolic discounting leads to inflation having significant long-run effects on real variables. JEL Classification: E20, E40, E50 Keywords: inflation, unemployment, Phillips curve, nominal inertia, monetary policy,

dynamic general equilibrium Corresponding author: Liam Graham Department of Economics University College London Gower Street London WC1E 6BT United Kingdom E-mail: Liam.Graham@ucl.ac.uk

1 Introduction

In this paper we show that if households have hyperbolic preferences and face

staggered nominal wage contracts, in�ation (via money growth) has signi�cant

long run e¤ects on output and employment. With our baseline calibration,

which takes the length of the contract period to be 1 year, a permanent increase

in in�ation of 1% is associated with an increase in output and employment of

approximately 0.2% for in�ation rates of up to around 10%. This is roughly of

the same order of magnitude as the empirical estimates of the long-run Phillips

curve in the well-known studies that have found a signi�cant tradeo¤, extending

from the early contributions of Phillips (1958) and Samuelson and Solow (1960)

to the recent work of Akerlof, Dickens and Perry (1996, 2000), Fair (2000) and

many others. Blanchard and Fisher (1989) wrote: �Most economists who came

to accept the view that there was no long-run trade-o¤ between in�ation and

unemployment were more a¤ected by a priori argument than by empirical evi-

dence.� We challenge the "a priori" arguments that the long-run Phillips curve

is necessarily vertical.

There are wide implications for macroeconomics and macroeconomic policy.

If money is no longer superneutral, unemployment cannot be decomposed into

cyclical and structural components; the concepts of an NRU or NAIRU need to

be reconsidered. A wide econometric literature identi�es supply and demand

shocks by a decomposition into permanent and transient e¤ects. This too be-

comes questionable if in�ation has permanent e¤ects on real variables. In terms

of policy, the presence of a non-vertical Phillips curve adds an extra degree of

complexity to the policymaker�s task.

Our model is a standard dynamic general equilibrium model with staggered

nominal contracts. Our only unconventional assumption is that households have

hyperbolic preferences. Instead of a constant rate of time preference, we model

households as showing a strong preference for a payo¤ today over a payo¤ to-

morrow, but a much weaker preference for a payo¤ in a year�s time over a payo¤

in a year and a day�s time. Our analysis here rests heavily on the seminal work

of Laibson (1996) and Barro (1999).

There are three channels by which in�ation has long-run e¤ects on real vari-

ables in our model:

(i) The employment cycling e¤ect : As in�ation increases, relative prices be-

come more volatile and thus the price allocation system works less e¢ ciently.

Our analysis captures this phenomenon through �uctuations in real wages, since

nominal wages are constant over the contract period while the price level rises

2

continually in an in�ationary steady state. The greater the in�ation rate, the

more volatile real wages and employment become. Since this employment cycling

(�rms substituting towards labor with relatively low real wages) is ine¢ cient, the

productivity of labor falls, reducing labor demanded and output supplied.

(ii) The labor smoothing e¤ect : When households have a preference for smooth-

ing their labor services through time, employment cycling increases their disu-

tility of labor, thereby inducing households to reduce their labor supply.

(iii) The discounting e¤ect : Time discounting reduces the weight of house-

holds�future disutility of labor. If nominal wages are constant over the contract

period, more in�ation means a greater fall in real wages, and a greater rise in

employment and the disutility of labor over the contract period. The more the

future disutility of labor is discounted, the less onerous labor becomes and the

more labor is supplied.

Whereas the �rst and second e¤ects generate a negative relation between

in�ation and macro activities (aggregate employment and output), the third

gives rise to a positive relation. We show that the discounting e¤ect dominates

for in�ation rates less than 10 percent.

That discounting leads to a non-vertical Phillips curve in new Keynesian

models is well-known1, however with the discount factor pinned down by the

real interest rate, the e¤ect remains small (Ascari, 1998, Graham and Snower,

2003). A model with hyperbolic discounting has agents discounting payo¤s in

the near future at a very high rate (discount factors of 60% or 70% are common

in the literature) while matching the observed real interest rate (Barro, 1999

refers to this as "observational equivalence"). It is these high discount factors

which apply to the wage setting decision, so our model signi�cantly ampli�es the

discounting e¤ect.

Our calibrations indicate that, with hyperbolic discounting, the discounting

e¤ect is dominant at in�ation rates less than 10%. Thus the relation between

in�ation (on the one hand) and employment and output (on the other) is signif-

icantly upward-sloping over this range.

The paper is organized as follows. Section 2 deals with preliminaries: the

relation of our analysis to the literature and a discussion of our central assump-

1Such a discounting e¤ect is present in the new Keynesian Phillips curve based on Calvopricing

�t = �Et�t+1 + �yt

which implies a relation between output and in�ation in the vicinity of the zero-in�ation steadystate

y =1� ��

�

3

tion: hyperbolic discounting. Section 3 presents our model and its calibration.

Section 4 gives some results along with sensitivity analysis to key parameters.

In section 5 we discuss the key assumptions of �xed contract length and no

indexation. Section 6 concludes.

2 Preliminaries

2.1 Relation to the Literature

Our analysis relates to the existing literature in two distinct ways. First, the

numerical results of our model are derived from calibrated parameters that are

standard in the New Phillips Curve models. Our only novel assumption in

this context is hyperbolic discounting, and our calibrations in that respect are

standard as well. Second, our results are in line with a growing body of empirical

macroeconomic evidence.

The existing empirical evidence on the slope of the long-run Phillips curve

has been mixed for some time, and has led major contributors such as Mankiw

(2001) to be "agnostic" on the issue. Given economists�predilection for the

classical dichotomy, it is striking how many empirical studies reject it. In recent

years, there has been a rapidly growing literature �nding empirical evidence for

a long-run tradeo¤ between in�ation and macroeconomic activity.

Ball (1997) shows that countries experiencing large and long declines in in-

�ation tend also to encounter comparatively large increases in their NAIRUs.

Ball (1999) suggests that such a relationship may be due to monetary policy:

countries with relatively contractionary policy in the 1980s tended to have rela-

tively large increases in their NAIRUs. Akerlof, Dickens and Perry (1996, 2000),

looking at data for the US �nd empirical evidence of a long-run tradeo¤ at low

in�ation rates. Lundborg and Sacklen (2001) apply a small open-economy ver-

sion of this model to Swedish data and �nd a negative long-run relationship

between in�ation and unemployment.

Dolado, López-Salido and Vega (2000) �nd some evidence of a permanent

tradeo¤ over the entire in�ation range for Spain using a data set running from

1964 - 1995. Fisher and Seater (1993) and Fair (2000) �nd long-run tradeo¤s as

well. King and Watson (1994) �nd that the estimated long-run trado¤ depends

on short-run identi�cation assumptions. Ericsson, Irons and Tryon (2001) �nd a

positive long-run relation between output and in�ation for most G-7 countries.

Gottschalk and Fritsche (2005) �nd a strong negative correlation between the

trend components of in�ation and unemployment for West Germany. Schreiber

and Wolters (2005) use a VAR cointegration analysis to �nd a negative long-run

4

relation between in�ation and unemployment for Germany, robust with respect

to speci�cation variations. For post-WWII data of the Canadian economy, Kous-

tas (1998) and Koustas and Voloce (1996) �nd that the evidence strongly rejects

the vertical long-run Phillips curve. Furthermore, using data covering the last

30-40 years for Austria, Denmark, Finland, France, Germany, Italy, Spain, Swe-

den and the UK, Koutas and Serletis (2003) present evidence against a vertical

long-run Phillips curve.

An important strand of the recent literature seeks to derive the Phillips

curve tradeo¤ from structural macro models, arguing that the traditional single-

equation models cannot fully capture the various important causal relations un-

derlying the Phillips curve. Bullard and Keating (1995) use a structural VAR to

estimate the long-run response of output to a permanent increase for a group of

16 countries. There point estimates of the response are positive for 10 countries

in the sample. Of these, the responses are signi�cantly di¤erent from zero at

a 90% con�dence interval for 5 countries (including the UK, Austria and Ger-

many), and at a 5% con�dence level for 2 more (Japan and Spain). Ahmed and

Rogers (1998) �nd long-run e¤ects of in�ation on consumption, investment and

output are positive, on the basis of 100 years of U.S. data.2 In the context of a

small structural rational expectations model, Coenen, Orphanides and Wieland

(2004) show how a zero bound on nominal interest rates a positive relation be-

tween in�ation and the output gap at low in�ation rates. Karanassou, Sala and

Snower (2003, 2005) �nd evidence of a signi�cant negative relation between in-

�ation and unemployment in the context of estimated labor market systems for

the U.S. and the E.U.

Our analysis contributes to a larger theoretical literature rationalizing a long-

run tradeo¤ between in�ation and macro activity. It is well-known that the

superneutrality result of Sidrauski (1967) depends on a number of strong as-

sumptions, such as the separability of consumption and leisure in utility. The

long-run e¤ect of money growth on macro activity can be shown to depend on

the nature of household preferences and the role of money in the economy (Or-

phanides and Solow (1990) provide a survey). This suggests that the slope of

the long-run Phillips curve is not necessarily vertical and its sign is an empirical

issue. Our rationale for the long-run Phillips curve, by contrast, does not rest

on non-separable utility functions, cash-in-advance, or money in the production

function.2Speci�cally, they estimate the e¤ects of exogenous changes in the long-run component of

in�ation on aggregate consumption, investment and output, in the context of a fully identi�edstructural vector error correction model.

5

Several authors (e.g. Ascari (2000, 2003) have shown that the Phillips curve

derived from New Keynesian microfoundations is highly nonlinear, with a pos-

itive relation between in�ation and macro activity at low in�ation rates and a

negative relation at high in�ation rates. In contrast to our analysis, however,

the discounting e¤ect is not strong in these models

In the context of a state-dependent menu cost model, Benabou and Konieczny

(1994) show how the long-run Phillips curve tradeo¤ depends on the asymme-

try of the pro�t function, the convexity of the product demand function, and

discounting. Konieczny (1990), Kuran (1986) and Naish (1986) provide non-

neutrality results in a broadly similar vein. Our model, by contrast, has time-

dependent nominal rigidities.

Our analysis is in some respects similar to the work of Akerlof, Dickens

and Perry (1996, 2000) who show that the Phillips curve becomes downward

sloping at in�ation rates below 3 - 4% when there are departures from rational

expectations or permanent nominal rigidities. In contrast, we show that the

long run tradeo¤ exists with rational expectations and only temporary nominal

rigidities. Our analysis is observationally distinct from that of Akerlof et al.

(1996), since our calibrations (below) imply a positive tradeo¤ between in�ation

and macroeconomic activity even when in�ation is substantially higher than 3-4

percent.

Further rationales for a long-run Phillips curve tradeo¤ are given by Holden

(2003) and Hughes-Hallett (2000). Holden (2003) shows how a non-superneutrality

of money can arise at low in�ation rates when the nominal wage can be changed

only by mutual consent in wage negotiations. Hughes-Hallett (2000) shows how a

long-run tradeo¤between in�ation and macroeconomic activity can arise through

aggregation over sectoral / regional Phillips curves with heterogenous short-run

slopes. Our analysis does not rely on such strategic considerations or aggregation

issues.

2.2 Hyperbolic discounting

The assumption of a constant rate of time preference is generally adopted for

analytical convenience, but a convincing underlying rationale for it has not been

articulated,3 and it is at odds with the available empirical evidence which sug-

gests that rates of time preference, viewed from the present moment, are high in

the short run but much lower in the long run.4 For example, people much prefer

3Axiomatic derivations are given by Koopmans (1960) and Fishburn and Rubinstein (1982).4See, for example, Ainslie (1992), Loewenstein and Thaler (1989), and Loewenstein and

Prelec (1992). Strotz (1956) was the �rst economist to study the implications of discount rates

6

$1000 today than that sum in a month from now, while their relative valuation

is much more evenly balanced when they have to choose between $1000 in a year

from now versus that sum in a year and a month. The observed time-varying

discount rates can be captured by hyperbolic discount functions, in which events

in � periods are discounted with factor (1 + ��)� =� ; where � and are positive

constants.5

When discount rates vary through time, households have a time - inconsis-

tency problem: households make their current decisions on the basis of prefer-

ences displaying low time discount rates for intertemporal choices lying in the

distant future; but once the future becomes the present, these households have

a much higher degree of impatience. Such time inconsistent preferences can be

modeled with standard game-theoretic tools as an intra-personal game, in which

today�s self is an independent player from each of the future selves. Today�s

self maximizes its utility, from today onwards into the future, knowing that the

future selves will make decisions based on di¤erent preferences. The decisions of

today�s self a¤ects the future selves; in particular, today�s consumption deter-

mines the wealth that will be bequeathed to the future selves. Also, the decisions

of the future selves a¤ect today�s self; in particular, future consumption a¤ects

the utility of today�s self, because today�s self maximizes its utility over the

present and future. The equilibrium of the intertemporal game played among a

household�s selves is a Nash equilibrium, based on this intertemporal consistency

of behaviors of the household�s selves.

There is a rapidly growing literature that uses time-varying discount rates

to explain a wide variety of other economic behaviors, including procrastination,

retirement saving, self-deception, self-control, etc.6.

A key feature of hyperbolic preferences is "observational equivalence" (Barro,

1999): in a model with hyperbolic discounting the e¤ective rate of time preference

(after solving for the Nash equilibrium of the household) is constant and the

discounting parameters can be chosen in such a way as to match observed values

of the real interest rate.

that decrease with the time horizon.5See Loewenstein and Prelec (1992). Simpler hyperbolic discount functions, such as 1=�

and 1= (1 + ��), are used in the psychology literature (e.g. Chung and Herrnstein (1961) andAinslie (1992)).

6See, for example, Akerlof (1991), Benabou and Tirole (2002, 2004), Carrillo and Marriotti(2000), Diamond and Koszegi (2003), and O�Donoghou and Rabin (1999a,b), Laibson, Repetto,and Tobacman (2001)

7

3 The Model

We present a simple dynamic general equilibrium model consisting of three types

of agent: households, �rms and a government.

There is a continuum of households which supply di¤erentiated labour to

�rms, consume goods, and hold money balances and bonds. Each household

sets its nominal wage, given the demand for its services. In the spirit of Taylor

(1979), we group households into N wage-setting cohorts, each of which sets a

nominal wage contract for N periods. Di¤erent cohorts set wages at di¤erent

times, uniformly staggered7.

Firms produce a homogeneous consumption good through di¤erentiated labour.

The government prints money and real bonds, and rebates the seigniorage pro-

ceeds to households as a lump sum.

3.1 Firms

The representative �rm uses all types of labour in a production function with a

constant elasticity of substitution (Dixit and Stiglitz (1977)) to produce a ho-

mogenous consumption good, and all labour types enter the production function

symmetrically: 8

yt =

24 1Zh=0

lt (h)��1� dh

35�

��1

(1)

where yt is output, lt (h) is the amount of labour chosen from household h, and

� is the elasticity of substitution between di¤erent labour types.

There is perfect competition in the product market. The �rm�s cost-minimization

7Another possibility, and one widely used in the literature, is to use Calvo contracts. How-ever for su¢ ciently high levels of money growth, Calvo contracts are not appropriate. Thereason is straightforward. With Calvo contracts, some households keep their nominal wageunchanged for a very long period of time, which means that, in the presence of in�ation, thereal value of this wage approaches zero. This implies that the �rm will wish to hire as muchof the labor of these households as possible, and as little of the other households. This isvery ine¢ cient so output approaches zero. Bakshi et al (2002) report that this happens within�ation of 5.5%; Ascari (2003) at somewhat higher levels.

8We use the following notational convention:nominal real

trended Xt xtdetrended Xt xtsteady state X x

8

implies a standard demand function for each household�s labour:

lt+i (h) =

�Wt (h)

Wt+i

���yt+i : 0 < i < N � 1; t = kN (2)

where W (h) is the wage set by household h and the aggregate wage index in

terms of e¢ ciency labour is

Wt =

24 1Zh=0

Wt (h)1�� dh

351

1��

(3)

Given that �rms are perfect competitors their markup will be zero so that the

real wage is constant and equal to unity W = P.

3.2 Households

Households consume, supply di¤erentiated labour and hold money balances and

bonds. A household h, which resets its wage in period t, faces a budget constraint

in that period given by:

ct (h) +Mt+1 (h)

Pt+Bt+1 (h)

Pt

= (1� � (h))Wt (h)

Ptlt (h) +

Tt (h)

Pt+Mt (h)

Pt+R

Bt (h)

Pt(4)

where ct is consumption, Wt the nominal contract wage, Mt nominal money

holdings, Pt the aggregate price index, R the gross real interest rate (assumed

constant) on bond holdings Bt; Tt net lump-sum transfers from government and

� (h) an income tax which we will allow to vary across households. The left-hand

side represents spending on consumption and closing real money balances and

bonds; the right hand side represents income from labour, pro�ts and opening

bond and money holdings net of taxes.

Household preferences are given by

Ut (h) = u (ct (h) ; lt (h) ;mt (h)) + 1Xi=1

�iu (ct+i (h) ; lt+i (h) ;mt+i (h)) (5)

where mt (h) =Mt(h)Pt

are the household�s real money balances.

Following Laibson (1996) and much of the subsequent literature, for analyt-

ical convenience we have approximated the hyperbolic discount function by a

"quasi-hyperbolic" discount function in which the discount factors from the per-

spective of the current period are 1; �; �2; �3::::. The discount factor between

9

the �rst two periods is �; but that between all subsequent periods is �. Thus

we can think of as a short-run discount factor, and � as a long-run discount

factor.

We choose a single-period utility function that has desirable long-run prop-

erties (see King, Plosser and Rebelo (1988)):

u (ct (h) ; lt (h) ;mt (h)) = log ct (h) + � l(1� lt (h))1�� � 1

1� � + �m logmt (h) (6)

where � l, �m, and � are positive constants: � is the elasticity of intertemporal

substitution of leisure. We assume the household writes nominal wage contracts

which last for N periods9.

When the discount rate is time-varying, the household�s behavior is time-

inconsistent: in future periods the household will have an incentive to change

plans made in the current period. Since household cannot commit itself to a

plan beyond the current period and since a household with rational expectations

will take this into account when making its decisions, it is standard practice

(e.g. Laibson, 1996) to formulate the household�s problem as a game played by

the household at time t (self t) against the household in future periods (future

selves).

In the appendix, we show that the unique equilibrium strategy st (h) in this

game of a household with preferences given by (5) facing constraints (4) and (2)

is a choice of consumption ct (h), real money balances mt (h), and, at the start

of each contract period, a choice of the nominal wage Wt (h) to be kept �xed for

the next N periods,

st (h) = fct (h) ;mt (h)g : t 6= kN (7)

st (h) = fct (h) ;W�t (h) ;mt (h)g : t = kN (8)

where the nominal reset wage (i.e. the nominal wage at the beginning of the

contract period) is given by

W�t (h) =

� l�

� � 1

(1� lt (h))�� lt (h) + N�1Pi=1

�i (1� lt+i (h))�� lt+i (h)

1��(h)ct(h)

N�1Pi=0

R�i lt+i(h)Pt+i

(9)

9We do not allow indexation within the contract period. We disucss the empirical plausi-bility of this assumption in section 5.

10

consumption is equal to the annuity value of after-tax lifetime wealth

ct (h) =R� 1R

(1� � (h))

24 1Xj=0

j(N�1)Xi=jN

R�iW�t (h)

Pt+ilt+i (h) + a0 (h)

35 (10)

and real money balances are given by

mt (h) =

�1

�mc�t (h)

�1� 1

R

PtPt+1

��� 1�

(11)

3.3 Aggregate quantities

The government�s budget constraint, in aggregate values,10 is

Bt+1Pt

+Mt+1

Pt+�tPt= R

BtPt+Mt

Pt� TtPt

(12)

where �t is any net revenue from the income tax.

A resource constraint closes the model.

Yt = Ct (13)

We de�ne aggregate labour as

lt =

1Zh=0

lt (h) dh (14)

3.4 Equilibrium

A competitive equilibrium for the above economy is a sequence of plans for

� allocations of households fc�t (h) ; lt (h) ;mt (h) ; bt (h)gh=1:1t=1:1

� prices fWtgt=1:1

� aggregate labour input fltgt=1:1

such that

1. Given prices, the allocations are Nash equilibria of the game-theoretic prob-

lem of each household (and hence solve the utility maximization problem)

10Note that the variables Bt;Mt; and Tt are not to be confused with the household�s holdingsBt (h) ;Mt (h) ; and Tt (h).

11

2. fWtgt=1:1 is the marginal product of aggregate labour

3. All markets clear

3.5 The steady state

For the remainder of the paper we consider the properties of a symmetric steady

state in which all household behave identically. We can then think of our

model as consisting of N agents, one from each cohort. Each agent faces an

identical problem so chooses the same wage (in real terms) when they can reset

it. This means the present value of labour income over the contract period

is the same for each agent, and (10) means that they each consume the same

constant proportion of their lifetime wealth in each period. Although the present

value of labour income over the contract is the same for every agent, the present

value of lifetime wealth is slightly di¤erent since at the start of time, di¤erent

cohorts start in di¤erent periods of their contracts with di¤erent incomes. We

can choose the income tax rate to equalise the present value of lifetime wealth

across cohorts, so all agents consume the same amount. Since each agent�s path

of income falls across the contract period, agents in di¤erent cohorts trade bonds

to smooth consumption11.

In the steady state, the money growth rate is constant and equal to the rate

of in�ation �:

� =Mt+1

Mt(15)

The equation for the aggregate wage index (3) implies the steady state real

contract wage is:

W �

P=

1

N

1� �N(��1)1� ���1

! 1��1

(16)

Combining (9) and (10) gives a labour supply relation for each household

(1� lt (h))�� lt (h) + N�1Xi=1

�i (1� lt+i (h))�� lt+i (h) =(� � 1) (R� 1)

� l�(17)

This sets the present value of the marginal disutility of labour with respect to

the wage (on the left-hand side) equal to the present value of the marginal utility

of the extra consumption resulting from a higher wage (on the right-hand side).

11Were we to model o¤-steady state behaviour, we would need to invoke some form of con-sumption insurance to deal with the consequent heterogeneity as in, for example, Huang, Liuand Phaneuf (2004).

12

Due to consumption smoothing, the right-hand side is constant since an increase

in the wage leads one-for-one to an increase in consumption and thus a fall in

its marginal utility.

Then (2), (16) and (17) fully describe the system and allow us to solve for

output and aggregate employment.

This steady state satis�es condition (1) for an equilibrium by construction.

Competitive factor markets ensure condition (2) holds and that the labour mar-

ket clears. Given consumption of all households is identical, it is straightforward

to show the goods market and bond market clears.

3.6 Calibration

We calibrate our model with standard values. For the elasticity of labour sub-

stitution, �, we choose a value of 5.12 We take the length of a contract to be

one year, following Taylor (1998), and assume it remains constant as in�ation

varies. We discuss the empirical rationale for this in detail in section 5. There

are two wage-setting cohorts (N = 2), so that every six months one cohort sets

its nominal wage and then keeps it �xed for a year. We take , the parameter

measuring the size of short-run subjective discounting to be 70%. Values of

between 60% and 70% are standard in the hyperbolic discounting literature, for

example Laibson (1996), based on empirical evidence such as Ainslie (1992). In

the appendix we show that the three time preference parameters �; and R are

related by

=1� �

� (R� 1) (18)

We set the annual real interest rate to 4%, which, along with our value for ,

ties down the household�s long-term discounting parameter, �, which is 0:986.We take households�preferences over leisure to be logs, � = 1. In Section 4.4

we analyze the sensitivity of our results to changes in all these variables.

We choose � l to be 2:85 which gives, from (17), steady state labour of 0.22

corresponding to 48 weeks of 40 hours worked per year. The value of �m af-

fects the level of real money balances held by the household but since utility is

separable in real money balances, this does not a¤ect our results.

4 Results

Figure 1 shows how aggregate steady state employment and output changes with

in�ation. Both employment and output increase monotonically with in�ation at

12Erceg et al (2000) use a value of 4; Huang and Liu (2002) use 6.

13

approximately the same rate in the long run, though the output line lies every-

where below the employment line (for reasons elucidated below). Over the range

of in�ation analyzed, the relation is close to linear, with a 1% increase in in�ation

associated with a rise of approximately 0.2% in output and employment.

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

2.00%

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

Inflation

ny

Figure 1: The steady state relation between real variables andin�ation

x-axis shows annualized in�ation; y-axis percent deviations of real variables from their

values at zero-in�ation

There are three channels by which in�ation a¤ects real variables in this model

which we call employment cycling, discounting and labour supply smoothing.

We consider these in turn to explain the form of the Phillips curve in �gure 1,

and how this form depends on our chosen parameter values.

4.1 Employment cycling

For the two wage-setting cohorts (N = 2), let cohort 1 set its wage at time t

while cohort 2 sets its wage at time t + 1 and each nominal contract wage is

set for two periods. Suppose that steady-state in�ation is positive. Since the

price level rises from period to period whereas each cohort�s nominal wage is

readjusted every second period, it follows that each cohort�s real wage is high

at the beginning of its contract period and low at the end of it. Speci�cally, at

time t, the real wage of cohort 1 is high and that of cohort 2 is low, and vice

versa at time t + 1. This is a simple way of capturing the empirical regularity

that as in�ation increases, relative prices become less stable.

14

Thus, the �rm at time t has a relatively low demand for cohort 1 and relatively

high demand for cohort 2, and vice versa at time t+1. This substitution towards

labour types with low real wages (and away from labour types with high real

wages) we call employment cycling.13. It is straightforward to show that the

degree of employment cycling is a function only of the in�ation rate and the

degree of substitutability between di¤erent labour types

l2l1= �� (19)

Since di¤erent labour types are imperfectly substitutable, employment cy-

cling is ine¢ cient in the sense that, for given aggregate employment l, more

employment cycling is associated with lower aggregate output. The greater is

steady-state in�ation rate, the more employment cycling �rms do, and thus the

lower the average productivity of labour. So output always increases by less than

employment as in�ation rises, and the output line lies below the employment line

in �gure 1. This captures the empirical regularity that as in�ation increases,

relative prices become less stable and this instability is a source of ine¢ ciency.

Employment cycling is familiar from the new Keynesian literature on optimal

monetary policy. For Woodford (2003) cycling between di¤erent types of labour

is the key distortion which leads to a stabilising role for monetary policy.

4.2 Labour smoothing

The labour-supply smoothing e¤ect occurs when � is positive. Due to employ-

ment cycling, individual households cannot provide a constant stream of labour

services through time. Since with nonzero � the marginal disutility of labour

rises with labour, these �uctuations in hours worked makes them worse o¤.

In response, they supply less labour at a given wage over the contract period.

Greater money growth leads to greater employment cycling, and thus lower aver-

age labour supply and, via the production function, lower output. In this way, the

labour-supply smoothing e¤ect weakens the steady-state employment-in�ation

tradeo¤ and, via the production function, the output-in�ation tradeo¤.

13At the levels of in�ation relevant to our recent macroeconomic experience, the degree ofemployment cycling is quite small: at 2% in�ation, employment �uctuates around its meanvalue by 2.5%. But at 10% in�ation this �uctuation rises to 16%. In practice, employmentcycling is of course much more likely to occur in form of hours variations than through hiringand �ring. There is evidence that overtime work is sensitive to the wage, and thus it seemsplausible that a limited degree of employment cycling may occur when a signi�cant degree ofin�ation occurs over the contract period.

15

4.3 Discounting

The discounting e¤ect may be understood intuitively through the following steps:

1. The labour supply relation (17), households set their contract wage so

that the present value over the contract period of the marginal disutility of

labor (MDL) with respect to the wage is equal to the present value of the

marginal utility of consumption (MUC) from the labor income generated

by a wage change.

2. When in�ation is positive, labor supply increases over the contract period

because of employment cycling and thus the MDL increases over the con-

tract period. By contrast, the MUC remains constant through time, due

to consumption smoothing.

3. Due to discounting, future marginal utilities and disutilities receive less

weight than current ones. Since the MDL increases over the contract

period whereas the MUC remains constant, an increase in discounting

reduces the present value of the MDL relative to the present value of the

MUC. In response, the household raises its labor supply thereby raising

the present value of the MDL to bring the two present values back into

equality. So, given in�ation, as the discount rate increases, labour supply

increases.

4. An increase in in�ation means employment cycling gets stronger so the

MDL increases more over the contract period and the more the discounting

e¤ect reduces the present value of the MDL relative to the present value

of the MUC. Consequently the household increases its labor supply to

bring the two present values back into equality. So, given discounting, as

the in�ation rate increases, labour supply increases.

Whereas the employment cycling e¤ect and the labor smoothing e¤ect gen-

erate an inverse relation between in�ation and macro activity, the discounting

e¤ect generates a positive relation. For the calibration above, as Fig. 1 shows,

the discounting e¤ect is dominant for in�ation rates less than 10 percent.

Barro (1999) showed that, in the absence of nominal contracting rigidities, an

economy with hyperbolic discounting is observationally equivalent to one with

exponential discounting in the sense that the equilibrium in an economy with

hyperbolic discounters coincides with that in an economy with exponential dis-

counters for a particular choice of the exponential discount factor. Our model

depends on this observational equivalence with respect to the real interest rate,

16

but observational equivalence is violated for other variables due to the existence

of multi-period nominal contracts. To see this, note that the denominator of

(9) is a �ow of (marginal) income so is discounted exponentially, whereas the

numerator is a �ow of marginal utility, which is discounted at the subjective

rate of time preference, i.e. hyperbolically. If contracts were only one period

long (N = 1) - as is implicitly assumed in models without nominal contract-

ing rigidities - neither the marginal disutility of labour nor the marginal utility

of consumption would be discounted. However when N > 1, the di¤erence

between the discounting of the denominator and the numerator breaks observa-

tional equivalence.

4.4 Sensitivities

For comparison, we �rst present the results of our model with standard expo-

nential discounting. Figure 2 shows the trade-o¤ between real variables and

in�ation for an exponential discount factor consistent with our choice of real

interest rate.

­0.50%

­0.40%

­0.30%

­0.20%

­0.10%

0.00%

0.10%

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

Inflation

ny

Figure 2: The steady state relation between real variables andin�ation with exponential discounting

x-axis shows annualized in�ation; y-axis percent deviations of real variables from their

values at zero-in�ation

With exponential discounting, the discounting e¤ect is very weak and the

other two e¤ects, both of which lead to a negative relation between output and

in�ation, dominate.

17

Table 1 shows how the strength of the discounting e¤ect, and hence the

extent of the monetary non-superneutrality, varies with the size of the discount

factor. The lower , the higher is the rate at which household�s discount the

future, so the stronger is the discounting e¤ect and the greater is the long-run

employment increase resulting from a given long-run rise in money growth. In

the hyperbolic discounting literature, values of 60% and 70% for the short-term

discounting parameter are common.

Table 1: Sensitivity to the short-run discount rate � increases from 2% to 3% � increases from 6% to 7%

Employment Output Employment Output

90% 0.05% 0.05% 0.02% 0.00%

80% 0.13% 0.12% 0.09% 0.07%

70% 0.21% 0.20% 0.17% 0.15%

60% 0.30% 0.29% 0.26% 0.24%

Though we have assumed that there are only two wage-setting cohorts (N =

2) to simplify our presentation, it is worth noting that using higher values has a

negligible e¤ect on our results.14

How does the tradeo¤ change with di¤erent values for �, the elasticity of

intertemporal substitution of leisure? Figure 3 shows these tradeo¤s for � =

2, the value used in Chari et al (1996) and for a higher value � = 5. This

contrasts with �gure 1 in two ways. Firstly, the tradeo¤ is weaker, the change in

employment if in�ation is increased from 0% to 5% is reduced by approximately

a third for � = 2 and a half for � = 5. Secondly, the curves become noticeably

non-linear: as in�ation increases, the degree of employment cycling increases and

so to does the strength of the labour-supply smoothing e¤ect. Thus the tradeo¤

gets weaker as in�ation increases.

14We have also made assumptions about technology and preferences. If we allow the pro-duction function to show diminishing returns to aggregate labour, the e¤ect on employmentis unchanged but output responds more weakly. Using a power function for labour with anexponent of 0.6 reduces the response of output in our baseline calibration by roughly one half.

18

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

1.6%

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

Inflation

Y (eta = 2)N (eta = 2)Y (eta = 5)N (eta = 5)

Figure 3: The steady state relation between real variables andin�ation: sensitivity to �

x-axis shows annualized in�ation; y-axis percent deviations of real variables from their

values at zero-in�ation

The parameter � measures the degree of substitutability between di¤erent

labour types in the �rm�s production function. Low � means that labour types

are poor substitutes; high � means that they are good substitutes. Increasing �

has three e¤ects. First, di¤erent labour types become closer substitutes and thus

a given amount of employment cycling is associated with higher output. In other

words, holding the degree of employment cycling constant, an increase in � raises

output. Second, an increase in � induces the �rm to raise employment cycling,

since it is now less costly to substitute among labour types over the contract

period. For a given level of aggregate employment, an increase in employment

cycling leads to a fall in output, since di¤erent labour types are imperfect substi-

tutes. For the above calibrations of the other parameters, it turns out that for

all but very low rates of in�ation, the second e¤ect dominates the �rst and the

output curve lies further below the employment curve as � increases. Thirdly,

the greater is the degree of cycling, the stronger is the discounting e¤ect so the

lower is the wage and the higher the level of employment and output for a given

level of in�ation.

Table 2 shows the e¤ects of a 1% increase in in�ation on output and em-

ployment. Because of the non-linear nature of the relation, we show the e¤ect

starting from two base values of 2% in�ation and 6% in�ation.

Table 2: Sensitivity to the elasticity of substitution between labour

19

types �Change in real variables from an increase in steady state in�ation

� increases from 2% to 3% � increases from 6% to 7%

� Employment Output Employment Output

1.01 0.04% 0.04% 0.04% 0.04%

2 0.09% 0.08% 0.08% 0.07%

5 0.21% 0.20% 0.17% 0.15%

10 0.37% 0.35% 0.23% 0.19%

The literature gives little guide as to what value of � is appropriate for ag-

gregate data. The elasticity of substitution between labour types de�ned by

occupations and seniority may be low in some instances, but economy-wide oc-

cupational unions are rare, while seniority-based unions are absent. In most

market economies nowadays, wage-setting cohorts usually comprise a wide rage

of occupations and seniority scales, so that the cohorts are presumably highly

substitutable for one another. Thus high values of �, e.g. values well above 5,

appear plausible in our analytical setting.

Table 3 shows how the slope of the tradeo¤ varies with the contract length.

As the contract length increases, our e¤ect becomes more pronounced, since the

degree of employment variation over the contract becomes greater. In response

to this, the labour supply smoothing e¤ect becomes stronger at higher levels of

in�ation, and so the relation becomes more non-linear.

20

Table 3: Sensitivity to the contract lengthChange in real variables from an increase in steady state in�ation

� increases from 2% to 3% � increases from 6% to 7%

Contract (years) Employment Output Employment Output

0.5 0.10% 0.10% 0.09% 0.09%

1 0.21% 0.20% 0.17% 0.15%

1.5 0.30% 0.28% 0.22% 0.17%

2 0.39% 0.36% 0.24% 0.17%

It is worth noting that our choice of parameters is based on estimates in the

literature, rather than derived from a relationship between output / employment

and in�ation that we estimate ourselves. Since this relationship can be estimated

in a wide variety of ways and is subject to considerable, ongoing controversy, we

deem it to be wiser - given our present state of knowledge - to present our results

for a wide range of plausible values for the relevant parameter values. This has

been done above and we have seen that a positive long-run relation between

in�ation and output/employment emerges over this wide range of values, at

moderate in�ation rates.

5 Discussion

In this section we discuss the validity of our underlying assumptions and draw out

some empirical implications of our model. Apart from hyperbolic discounting,

the evidence for which we have discussed above, our central results rest on two

key assumptions: that wages are set annually in nominal terms, and that within

this contract period of one year there is no indexation.

When thinking about the nature of wage contracts it is important to distin-

guish between unionised and non-unionised workers. Taylor (1998), reviewing

the direct empirical evidence for the unionised sector in the US, cites a large

number of studies that suggests that annual contracts are the most common

length of wage setting interval. The wide variation across countries is captured

by Groth and Johansson (2004) who update the study of Bruno and Sachs (1985)

with data from 1985 - 1995. They assign countries an index which takes a value

of 0 if the average length of union contract is a year or less, 1 if it is from 1 - 3

years and 2 if it is greater than 3 years. The mean of this index across OECD

countries was 0.94 for 1985 - 1995, suggesting a mean contract length greater

than a year.

There is much less evidence on the non-unionised sector, partly because the

most common source of data are panel data sets (the PSID for the US, or the

21

BHPS for the UK) which typically collect their data annually so necessarily

miss any sub-annual wage setting. However there does seem a consensus in the

literature that most wages are set annually. For example, Taylor (1998), for

the US, writes that "setting nominal wages at a �xed level for more than several

quarters and usually for as long as a year appears to be just as prevalent for

workers who are not in unions, or do not have formal employment contracts,

as for union workers with employment contracts". Smith (2000), referring to

the UK, notes that "pay negotiation in the United Kingdom typically occur

at annual intervals, and pay awards are often made every 12 months even for

workers who are not covered by collective bargaining" and Brown et al (2004)

state that the annual wage settlement is "the principal source of pay change for

most employees in Britain". This is consonant with estimates of the degree of

wage stickiness (reviewed in Taylor, 1998)) using aggregate data which typically

�nd the average length of time between wage changes to be greater than one

year.

Because we assume the length of nominal contracts is �xed at one year while

we vary in�ation from 0% to 10% our model is subject to a version of the Lucas

critique. Wouldn�t agents change their wage setting behaviour as steady state

in�ation varies? So an important empirical question is the degree to which

the contract period changes with steady state in�ation. Taylor (1998) writes

than the average contract period fell to one year (our baseline value) during the

great in�ation of the 1970s, when average in�ation was well above the range we

consider in this paper. Looking across the OECD, Groth and Johansson (2004)

�nd their index variable was 0.78 in the period 1975 - 85, still consistent with an

average contract length above a year, and rose to 0.94 in the next ten years when

average in�ation was lower. So this suggests that modelling nominal contracts

as �xed at one year over a range of in�ation from 0 - 10% as we do in this paper

is, if anything, a conservative assumption and we could expect contracts to be

longer now average in�ation is low.

Can we understand this surprising rigidity of the contract length theoreti-

cally? We have analyzed this case extensively in previous work (Graham and

Snower, 2004). There, rather than treating the contract period as exogenous,

we allow households to choose the length of their wage contract period, assum-

ing that there is a �xed cost to changing wages. To calibrate this �xed cost we

assume that wages are set for one year at an in�ation rate of 5%, on the basis

that this is the average level of US in�ation over the past 20 years. We �nd

the �xed cost is large. This should not be very surprising since it represents

much more than a negotiation cost (e.g. the cost of the time spent negotiating,

22

the expected cost of a breakdown in negotiations, etc.), as wage adjustments are

typically accompanied by performance and salary reviews.

Because this cost is large, the e¤ect of endogenising the contract length was

minimal over a range of 0% - 10% in�ation, though as in�ation increases above

10% the e¤ect may become rapidly larger. Further, the e¤ect of endogenisation

is to strengthen the tradeo¤ between in�ation and real variables for rates of

in�ation less than the reference level of 5%. Since the costs of in�ation decrease

with in�ation, the household is less willing to pay the �xed cost so the contract

period lengthens and the tradeo¤s become stronger.

Our assumption of Taylor contracts, without intra-contract indexation, in-

stead of Fisher contracts, is similarly open to the Lucas critique. First note

that our model of annual nominal contracts nests annual indexation since wages

are optimally reset each year. Only indexation within the annual contract pe-

riod would a¤ect our results. The empirical evidence on indexation is striking,

Card (1986), writing about a period when average in�ation was well over the

top of the range we consider, writes that "perhaps no more than 10% of all

US workers are covered by cost-of-living provisions". This surprising result is

because indexation is only observed in union contracts, and, as Card (1983) re-

ports, only in around 60% of these, and "escalation provisions are rare in the

non-union sector". Further, while one can �nd evidence of quarterly COLA�s,

the vast majority seem to be annual (Kaufman and Woglom, 1986, Vroman,

1985). More recently, Christo�des and Leung (2003), looking at Canadian data

from 1976 - 1999, write that "very few contracts contain COLA clauses".

So while wage indexation is a feature of very high in�ation rates (Marinakis,

1997) it seems largely absent over the range of in�ation we consider. The

decision of whether to index or not is very similar to the choice of contract length,

and most models that attempt to explain why indexation is not observed (e.g.

Ball (1988), and Calmfors and Johansson (2002) for a general review) assume

indexation has a �xed cost. Exceptions are Danziger (1988) and Mukerji and

Tallon (2004) who show that, if agents are uncertainty averse, they will choose

not to index wage contracts even if there are no �xed costs of doing so.

6 Concluding comments

We have shown that the interaction of staggered wage contracts with hyperbolic

discounting leads to in�ation having signi�cant long run e¤ects on real variables.

What are the implications for monetary policy and macroeconomic analysis? We

address these in turn.

23

As to the optimal in�ation rate, �rst note that output at zero in�ation is not

�rst-best optimal due to imperfect competition. As in�ation increases, output

(and hence consumption) and employment increase, and real money balances

fall (as a consequence of the in�ation tax). The �rst increases the utility of

households; the second and third decrease it. As the nominal interest rate

tends to zero, the third e¤ect dominates giving the Friedman rule. For positive

rates of in�ation, the relative magnitude of these three e¤ects will depend on

the calibration of the model. We leave a detailed analysis of implications of our

model for monetary policy for future work.

According to the new Neoclassical Synthesis, unemployment can be divided

into structural and cyclical components, where structural unemployment or the

NAIRU depends on supply-side factors such as productivity growth and the de-

gree of imperfect competition, while cyclical unemployment depends on demand-

side factors such as those associated with monetary policy swings. If we de�ne

the Phillips curve as the relation between in�ation and real macro activity that

results from changes in monetary policy, then the new Neoclassical Synthesis

implies that the Phillips curve tradeo¤ exists in the short-run but not in the

long-run. This approach suggested that demand- and supply-side shocks could

be distinguished econometrically by examining whether they were short- or long-

lived, see, for example, Blanchard and Quah (1989). Our analysis calls these

views into question. If monetary policy can have long-term e¤ects on real vari-

ables, unemployment cannot be decomposed into cyclical and structural compo-

nents; the concept of a NAIRU needs to be reconsidered.

24

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